[Page E-10 ( continued )] | 1. | The Army is conducting war games in Europe. One simulated encounter is between the Blue and Red Divisions. The Blue Division is on the offensive; the Red Division holds a defensive position. The results of the war game are measured in terms of troop losses. The following payoff table shows Red Division troop losses for each battle strategy available to each division: | | Red Division Strategy | | Blue Division Strategy | A | B | C | | 1 | 1,800 | 2,000 | 1,700 | | 2 | 2,300 | 900 | 1,600 | Determine the optimal strategies for both divisions and the number of troop losses the Red Division can expect to suffer. | | 2. | The Baseball Players' Association has voted to go on strike if a settlement is not reached with the owners within the next month. The players' representative, Melvin Mulehead, has two strategies (containing different free agent rules, pension formulas, etc.); the owners' representative, Roy Stonewall, has three counterproposals. The financial gains, in millions of dollars, from each player strategy, given each owner strategy, are shown in the following payoff table:
[Page E-11] | | Owner Strategy | | Player Strategy | A | B | C | | 1 | $15 | $9 | $11 | | 2 | 7 | 20 | 12 | -
Determine the initial strategy for the players and for the owners. -
Is this a pure or a mixed strategy game? Explain. | | 3. | Mary Washington is the incumbent congresswoman for a district in New Mexico, and Franklin Truman is her opponent in the upcoming election. Because Truman is seeking to unseat Washington, he is on the offensive, and she hopes to minimize his gains in the polls . The following payoff table shows the possible percentage point gains for Truman, given the political strategies available to each politician: | | Mary Washington Strategy | | Franklin Truman | A | B | | 1 | 7 | 3 | | 2 | 6 | 10 | -
Determine the optimal political strategy for each politician and the percentage gain in the polls Franklin Truman can expect. -
Solve this problem by using the computer. | | 4. | Edgar Allan Melville is a successful novelist who is negotiating a contract for a new novel with his publisher, Potboiler Books, Inc. The novelist's contract strategies encompass various proposals for royalties, movie rights, advances, and the like. The following payoff table shows the financial gains for the novelist from each contract strategy: | | Publisher Strategy | | Novelist Strategy | A | B | C | | 1 | $80,000 | $120,000 | $90,000 | | 2 | 130,000 | 90,000 | 80,000 | | 3 | 110,000 | 140,000 | 100,000 | -
Does this payoff table contain any dominant strategies? -
Determine the strategy for the novelist and the publisher and the gains and losses for each. | | 5. | Two major soft drink companies are located in the Southeastthe Cooler Cola Company and Smoothie Soft Drinks, Inc. Cooler Cola is the market leader, and Smoothie has developed several marketing strategies to gain a larger percentage of the market now belonging to Cooler Cola. The following payoff table shows the gains for Smoothie and the losses for Cooler, given the strategies of each company:
[Page E-12] | | Cooler Cola Strategy | | Smoothie Strategy | A | B | C | | 1 | 10 | 9 | 3 | | 2 | 4 | 7 | 5 | | 3 | 6 | 8 | 4 | Determine the mixed strategy for each company and the expected market share gains for Smoothie and losses for Cooler Cola. | | 6. | Tech is playing State in a basketball game. Tech employs two basic offensesthe shuffle and the overload; State uses three defensesthe zone, the man-to-man. The points Tech expects to score (estimated from past games), using each offense against each State defense, are given in the following payoff table: | | State Defense | | Tech Offense | Zone | Man-to-Man | Combination | | Shuffle | 72 | 60 | 83 | | Overload | 58 | 91 | 72 | Determine the mixed strategy for each team and the points Tech can expect to score. Interpret the strategy probabilities. | |